One of the few surviving players to have appeared in every season of Celtic Rugby/PRO12 tournaments since 2001/02, Swift took his tally to 145 matches last season.
Now the powerhouse lock needs just five more appearances to become the first to reach 150 league caps.
Connacht have five games in September and there is every likelihood that Swift, who turns 35 on October 18, could reach the landmark by the end of next month after signing a year extension to his contract at the province.
Hot on his heels in the race to 150 appearances are Cardiff Blues out-half Ceri Sweeney (139), Swift's Connacht team-mate John Muldoon (136), Ospreys prop Adam Jones (135), new Connacht recruit Dan Parks (132) and Glasgow Warriors skipper Ali Kellock (132).
Meanwhile, the race is on to become the first player to score 50 tries in the RaboDirect PRO12 - and it could be either Tommy Bowe or Tim Visser.
No one would have bet against Bowe, with 46 tries in 116 games, hitting the half century mark in recent years, but the arrival of the 'flying Dutchman' at Edinburgh has dramatically changed the picture.
Visser has notched a very impressive 37 tries in 57 appearances for Edinburgh in his first three seasons in the competition and is scoring at a rate of one try every 1.54 games. His 14 tries in the 2010/11 campaign eclipsed Bowe's old record for a season and he followed that up with 13 more last season.
Bowe, back on home soil at Ravenhill this season, notched his first tournament try back in the 2003/04 season. The Ireland and Lions winger has 28 tries for Ulster and 18 with the Ospreys.
He needs just four more to reach the half century mark, but his rate of scoring has eased to one per 2.5 games.